Parallel and Distributed Methods for Constrained Nonconvex Optimization—Part I: Theory

Abstract

Abstract-In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex (smooth) objective function, subject to nonconvex constraints, based on inner convex approximations. This Part II is devoted to the application of the framework to some resource allocation problems in communication networks. In particular, we consider two non-trivial case-study applications, namely: (generalizations of) i) the rate profile maximization in MIMO interference broadcast networks; and the ii) the max-min fair multicast multigroup beamforming problem in a multi-cell environment. We develop a new class of algorithms enjoying the following distinctive features: i) they are distributed across the base stations (with limited signaling) and lead to subproblems whose solutions are computable in closed form; and ii) differently from current relaxation-based schemes (e.g., semidefinite relaxation), they are proved to always converge to d-stationary solutions of the aforementioned class of nonconvex problems. Numerical results show that the proposed (distributed) schemes achieve larger worstcase rates (resp. signal-to-noise interference ratios) than state-ofthe-art centralized ones while having comparable computational complexity.

Transmit beamformer and quantization design for multi-carrier CRAN CoMP

“…The constraint (8a) is an under-estimator for γ k,n and b k,n in (8b) is a over-estimator for the total interference seen by user k on subchannel n. Even after relaxing (2) with (8), the resulting problem is not convex due to the constraints (8a) and (7f). Therefore, we adopt successive convex approximation (SCA) method in [13], [14], where the nonconvex constraints are replaced by a sequence of convex subsets that can be solved iteratively until convergence.…”

Distributed Optimization Methods for Nonconvex Problems with Inequality Constraints over Time-Varying Networks

“…Until recently, there are some references designing algorithms for distributed nonconvex problems, such as [16,17]. In [16], Lorenzo and Scutari developed a novel distributed algorithm for solving unconstrained nonconvex optimization problems which associated a multiagent network with timevarying connectivity.…”

“…They combined a successive convex approximation technique and a dynamic consensus mechanism to realize distributed computation as well as local information exchange in the network. For nonconvex optimization problems with constraints, Scutari et al [17] proposed a successive convex approximation method by solving a sequence of strongly convex subproblems while maintaining feasibility. They have shown that their proposed method can converge to a stationary solution of the original constrained nonconvex problem under consideration.…”

“…They have shown that their proposed method can converge to a stationary solution of the original constrained nonconvex problem under consideration. However, the method proposed in [17] is not suitable in the distributed setting. Additionally, the Lagrange dual method is utilized to solve the sequence of strongly convex subproblems, which may largely enlarge the dimension of the problem due to introducing the dual variables.…”

“…The work in this paper is closely related to the previous works [16][17][18][19]. Our method is based on the algorithm proposed in [16], but the problem we consider in this paper is different from the one in [16], since we investigate a nonconvex problem with global inequality constraints.…”